A better conditioned Domain Wall Operator

0.09 MB·English

by  Hartmut Neff

A better conditioned Domain Wall Operator
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1 H. Neff, Luzernerstrasse 43, 6330 Cham, Switzerland ∗
0
2

n June 6, 2015
a
J

1
2
Abstract

]
t
a AvariationoftheDomainWalloperatorwithanadditionalparam-
l eter α will be introduced. The conditioning of the new Domain Wall
-
p operatordependsonα,whereasthecorresponding4Dpropagatordoes
e not. The new and the conventional Domain Wall operator agree for
h
α = 1. By tuning α, speed ups of the linear system solvers of around
[
20% could be achieved.

2
v
0 1 Introduction
5
9
4 A variation of the Domain Wall operator is suggested here. It introduces a
0 parameter α that appears only as a global factor in the 4D matrix elements.
.
1 Therefore, this generalization is simple in structure and the Domain Wall
0
formalismandthereductiontothe4DOverlapformalismcanbeusedalmost
5
1 unchanged. Details about the Domain Wall and the Overlap formalism and
: how they can be translated into each other can be found here [1, 2, 3, 4, 5,
v
i 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. As a reference for notation and for
X
the sake of completeness the standard 5D to 4D reduction will be rederived
r
a in appendix A and B.
∗Email: hartmutneff@aol.com
1
2 The better conditioned Domain Wall operator
The new Domain Wall operator introduces an additional parameter α,
D (m) =
α
D (P +αP ) αD P 0 ··· −mD P
1+ − + 1− − 1− +
 αD2−P+ αD2+ αD2−P− ··· 0 
0 αD P αD ··· ··· (1)
3− + 3+
 
 ··· ··· ··· ··· αD P 
 (Ls−1)− − 
 −mD P 0 ··· αD P D (P +αP )
 Ls− − Ls− + Ls+ + − 
with
D = b D +1, D = c D −1, (2)
i+ i w i− i w
P = 1(1+γ ), P = 1(1−γ ). (3)
+ 2 5 − 2 5
D denotes the Wilson Dirac matrix
w
1
D (M )=(4+M )δ − (1−γ )U (x)δ +(1+γ )U†(y)δ . (4)
w 5 5 x,y 2 µ µ x+µ,y µ µ x,y+µ
(cid:2) (cid:3)
Multiplying eq.(1) from the right with P, (see eq.(14)), leads to
P P 0 ··· 0
− +
 0 P− P+ ··· 0 
D P = D 0 0 P ··· ··· (5)
α α −
 
 ··· ··· ··· ··· P 
 + 
 P 0 ··· 0 P 
 + − 
Q c αQ 0 ··· 0
1− − 1+
 0 αQ2− αQ2+ ··· 0 
= γ 0 0 αQ ··· ··· (6)
5 3−
 
 ··· ··· ··· ··· αQ 
 Ls−1+ 
 Q c 0 ··· 0 αQ 
 Ls+ + Ls− 
1 0 0 ··· 0
 0 α 0 ··· 0 
= D P 0 0 α ··· ··· ≡ D PA. (7)
1 1
 
 ··· ··· ··· ··· 0 
 
 0 0 ··· 0 α 
 
To find the 4D propagator, eq.(38) has to be solved,
D (m)P~y = D (1)P~b, (8)
1 1
2
withsource~band4Dpropagatory . Theindependenceofthe4Dpropagator
1
from α follows directly,
D (1)P~b = D (m)P~y = D (m)PA−1~y = D (m)P~z , (9)
1 1 α α
with A~z =~y and therefore z = y .
1 1
3 Results
Inthissection,theαdependenceoftheconditioningofD willbepresented.
α
The computations were done on 3 MILC gauge fields of size 163×32, down-
loaded at NERSC. The conjugate gradient method on the normal equation
was used to solve eq.(9).
The red black preconditioned version of D was used in the form,
α
D = 1 −I−1D I−1D . (10)
bb bb bb br rr rb
This version of red black preconditioning allows for an efficient use of the
Zolotarev approximation to the sign function. This is contrary to what has
been said in [17], where we used the matrix,
D = I −D I−1D , (11)
bb bb br rr rb
instead. This is due to the fact that the rows of eq.(11) with large Zolotarev
coefficients cause the convergence to slow down. This behaviour can be
improved by scaling all rows that contain a Zolotarev coefficient larger than
one with a factor equal to the inverse of the Zolotarev coefficient. This can
be seen as a preconditioning from the left. But the even better method is to
take eq.(10) where the preconditioning from the left cancels out and where
the weighting of the rows is done automatically.
The same behaviour can be observed for M¨obius coefficients b and c
i i
larger than one.
Let n (α) be the number of iterations for the residual to be of the order
i
of O(−8), where i runs over the color and Dirac source indices and over
the three gauge fields. The graphs in this section show the relative count
n (α)/n (α = 1), together with the standard deviation, for a series of α
i i
values.
For theremainingparameters, thequarkmass, the5th dimensionL and
s
the M¨obius coefficients b and c , the optimal alpha values and speed ups
i i
are summarised in the following table.
3
b=1.0, c=1.0, m=0.06, L =4
s
1
nt 0.95
u
o
c
n 0.9
o
erati 0.85
e it
v 0.8
ati
el
r 0.75
0.7
0.5 0.6 0.7 0.8 0.9 1
α
Figure 1: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
Mass L b , c Best α Speed Up
s i i
0.06 4 1, 1 0.55 25%
0.06 6 1, 1 0.55 24%
0.06 8 1, 1 0.55 22%
0.06 10 1, 1 0.6 19%
0.06 12 1, 1 0.6 17%
0.01 8 1, 1 0.55 23%
0.06 8 1.7, 0.7 0.6 20%
0.06 10 Zolotarev 0.4 17%
Acknowledgment: I thank Richard Brower and Kostas Orginos for dis-
cussionsandTonyKennedyfordiscussionsandthecodetocomputeZolotarev
coefficients.
4
b=1.0, c=1.0, m=0.06, L =6
s
1
nt 0.95
u
o
c
n
o 0.9
ati
er
e it 0.85
v
ati
el 0.8
r
0.75
0.5 0.6 0.7 0.8 0.9 1
α
Figure 2: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
b=1.0, c=1.0, m=0.06, L =8
s
1
nt 0.95
u
o
c
n
o 0.9
ati
er
e it 0.85
v
ati
el 0.8
r
0.75
0.5 0.6 0.7 0.8 0.9 1
α
Figure 3: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
5
b=1.0, c=1.0, m=0.06, L =10
s
1
nt 0.95
u
o
c
n
o 0.9
ati
er
e it 0.85
v
ati
el 0.8
r
0.75
0.5 0.6 0.7 0.8 0.9 1
α
Figure 4: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
b=1.0, c=1.0, m=0.06, L =12
s
1
0.98
unt 0.96
o
c 0.94
n
o 0.92
ati
er 0.9
e it 0.88
v
ati 0.86
el 0.84
r
0.82
0.8
0.5 0.6 0.7 0.8 0.9 1
α
Figure 5: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
6
b=1.0, c=1.0, m=0.01, L =8
s
1
nt 0.95
u
o
c
n
o 0.9
ati
er
e it 0.85
v
ati
el 0.8
r
0.75
0.5 0.6 0.7 0.8 0.9 1
α
Figure 6: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
b=1.7, c=0.7, m=0.06, L =8
s
1
nt 0.95
u
o
c
n
o 0.9
ati
er
e it 0.85
v
ati
el 0.8
r
0.75
0.5 0.6 0.7 0.8 0.9 1
α
Figure 7: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32.
7
b=1.0, c=1.0, m=0.06, L =10, Zolotarev
s
1
nt
u
o 0.95
c
n
o
ati
er 0.9
e it
v
ati 0.85
el
r
0.8
0.4 0.5 0.6 0.7 0.8 0.9 1
α
Figure 8: Relative iteration count n (α)/n (α = 1) for 3 gauge fields of size
i i
163 ×32, with b = c .
i i
References
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A Domain Wall to Overlap transformation
To keep notation simple, we perform the transformation with L = 4 sites
s
in the 5th dimension. A generalisation to arbitrary L is straightforward.
s
The Domain Wall to Overlap transformation reads,
LD (m)R = FD5 (m). (12)
DW OV
The transformation matrices take the form (for L sites in the 5th dimen-
s
sion),
F = LD (1)R, (13)
DW
and
1 S S S S S S Q−1 0 0 0
1 1 2 1 2 3 1−
L=L L = 0 1 S2 S2S3  0 Q−2−1 0 0 γ ,
1 2 0 0 1 S 0 0 Q−1 0 5
 3  3− 
 0 0 0 1  0 0 0 Q−1 
  4− 
P P 0 0 −1 0 0 0
− +
R = PR =  0 P− P+ 0  −S2S3S4c+ 1 0 0 ,
1 0 0 P P −S S c 0 1 0
− + 3 4 +
  
 P 0 0 P  −S c 0 0 1 
+ − 4 +
  
D4 (m) 0 0 0
OV
 0 1 0 0 
D5 (m)= . (14)
OV 0 0 1 0
 
 0 0 0 1 
 
The matrix entries are defined as follows,
Q = γ D (b P +c P )+1, Q = γ D (b P +c P )−1,
i+ 5 w i + i − i− 5 w i − i +
S = T−1 = −Q−1Q ,
i i i− i+
c = P −mP , c = P −mP . (15)
+ + − − − +
T−1 is called the transfer matrix.
i
The matrix multiplications will be performed in the following order,
L L D (m)PR = L L M R = L M R = L M = M . (16)
1 2 DW 1 1 2 1 1 1 2 1 1 3 4
10